Where do 'random seeds' get used in deep neural networks?
I suppose the most common part where it will be used is in the initialization of weights before training; the best ways currently known to do that involve randomness.

If you use Dropout during training (randomly setting some activation levels to zero to combat overfitting), that also involves randomness, so your seed could also have influence there. Dropout should not be used anymore after training, although it could if you accidentally implement it to be used there. If you don't make that mistake though, your seed here should also only matter during training.

Depending on implementation, I suppose the seed could also have influence on random ordering of input data between epochs / random selection of minibatches from the training dataset during training. Of course, this is very much implementation-dependent. If you end up implement that kind of data processing yourself and don't do it through some framework, you'll also be the one who determines what random seed has influence on that process.

In general (barring any special cases that I'm unaware of), a Neural Network should behave deterministically after training; if you give it the same input, it should provide the same output, your random seed should no longer have influence after training.